Consider the following set of three​ distributions, all of which are drawn to the same scale. Identify the two distributions that are normal. Of the two normal​ distributions, which one has the larger​ variation?

xy

Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins just above the origin, rises gradually to a maximum and falls gradually to just above the x-axis.

xy

Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins at the origin, rises sharply to a maximum and falls gradually to just above the x-axis.

xy

Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins at the origin, rises sharply to a maximum and falls sharply to the x-axis.

​(a)

​(b)

​(c)

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Part 1

The two normal distributions are 

▼
 
(a) and (b)

(b) and (c)

(a) and (c)

​, 

where 

▼
 
(c)

(b)

(a)
 
has the larger standard deviation.

In this problem, we are asked to identify which of the three distributions are normal and which of the two normal distributions has a larger variation (or standard deviation).

### Step 1: Identifying Normal Distributions
A normal distribution is characterized by a smooth, bell-shaped curve that is symmetric around its peak. Let's analyze the descriptions of the three curves:

- **Curve (a):** A smooth curve begins just above the origin, rises gradually to a maximum, and falls gradually to just above the x-axis.  
  - **Observation:** This sounds like a bell-shaped, symmetric curve. Therefore, **(a) is a normal distribution**.
  
- **Curve (b):** A smooth curve begins at the origin, rises sharply to a maximum, and falls gradually to just above the x-axis.  
  - **Observation:** This is also a bell-shaped curve, though with one side rising more steeply than the other. Despite the asymmetry in how it rises and falls, the description suggests a bell-shaped curve, so **(b) is also a normal distribution**.

- **Curve (c):** A smooth curve begins at the origin, rises sharply to a maximum, and falls sharply to the x-axis.  
  - **Observation:** This is not a normal distribution. The rise and fall are both steep, which resembles a non-symmetric distribution such as a triangular or other skewed distribution. Therefore, **(c) is not a normal distribution**.

Thus, the two normal distributions are **(a) and (b)**.

### Step 2: Comparing Variation
The standard deviation (or variation) of a distribution indicates the spread of the data. A curve with a larger spread will have a larger standard deviation. We can compare the descriptions of curves (a) and (b):

- **Curve (a):** Rises gradually and falls gradually.
- **Curve (b):** Rises sharply but falls gradually.

Since curve (a) rises and falls more gradually, it has a wider spread, implying a **larger standard deviation** compared to curve (b), which rises more sharply.

### Conclusion:
- The two normal distributions are **(a) and (b)**.
- Curve **(a)** has the larger standard deviation.

Would you like more details or have any questions? Here are 5 related questions to explore further:

1. How is the standard deviation of a normal distribution calculated?
2. What are the key properties of a normal distribution?
3. How can we distinguish between normal and non-normal distributions in real data?
4. How does the shape of a curve affect the interpretation of its standard deviation?
5. What are examples of real-world phenomena that follow normal distributions?

**Tip:** In a normal distribution, about 68% of the data falls within one standard deviation of the mean, and 95% falls within two standard deviations.

Consider the following set of three distributions, all of which are drawn to the same scale. Identify the two distributions that are normal. Of the two normal distributions, which one has the larger variation?

xy

Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins just above the origin, rises gradually to a maximum and falls gradually to just above the x-axis.

xy

Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins at the origin, rises sharply to a maximum and falls gradually to just above the x-axis.

xy

Quadrant one of a coordinate plane has a horizontal x-axis and a vertical y-axis. A smooth curve begins at the origin, rises sharply to a maximum and falls sharply to the x-axis.

Learn how to identify normal distributions from a set of curves and compare their standard deviations. This example problem explains the characteristics of normal distributions and shows how gradual vs sharp changes in curves affect the spread. Suitable for high school students in Grades 9-12.

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